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Full text of "Treatise On Analysis Vol-Ii"

8 LEBESGUE'S CONVERGENCE THEOREMS 125 The function/' is 7c(/^)-integrable if and only if/' o n is /*-integrable, and in that case we have If d(7t(uj) = \ (f o n) dp. 8. LEBESGUE'S CONVERGENCE THEOREMS (13.8.1) Let (/,) be an increasing sequence of integrable functions. For sup/, to be integrable, it is necessary and sufficient that sup \fn d^< + 00, » n J and if this condition is satisfied we have (13.8.11) = T \fn ^ + (-/.) dp = Since f* fn d\i > — oo for all w, we have f * (sup/,) dp = sup \fndfji J J \ n i n J (13.5.7), This already shows that the condition is necessary. Conversely, if the condition is satisfied, let /= sup/, . Then, given any e > 0, there exists n n such that the function/—/, (which is defined almost everywhere because/ and/, are finite almost everywhere (13.6.4)) satisfies J*(/-/n) dp ^ J/dAi + J*(- (13.5.6). But there exists a function ue «pfR(X) such that \\fn — u\ dp ^ s (13.7.2), hence by (13.5.6) we have [V- "I dp £ r\f-fn\ dll -I- f*^ - Il| d|X ^ 26, and the result follows by (13.7.2). There is of course a corresponding theorem for decreasing sequences of integrable functions, obtained by replacing/, by ~/rt in (13.8.1). (1 3.8.2) Let (/,) be any sequence of integrable functions. In order thatf— sup/, n should be integrable, it is necessary and sufficient that there should exist a function g^Q such that | g dp < +00 andfn ^ g almost everywhere. The condition is obviously necessary (take g =/+). Conversely, if it is satisfied, let gn = sup /fc. Then gn is integrable (13.7.4) and/=sup#w.tion on X' with values in R, we have